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George H, you wrote:
"And now, in real life, we have to combine those two motions together, the
symmetrical parabola with the steadily changing altitude, increasing at 10
minutes an hour. And I suspect that the only way to convince Bill will be to
persuade him to take a bit of graph paper and combine them for himself."

Oh my...
George, I see that you're still confused by the language here. So let me be
exceedingly blunt: NO ONE HAS SUGGESTED THAT THE CURVE IS NO LONGER A
PARABOLA. Of course it is, by mathematical necessity it is. The WHOLE POINT
is that the curve is SHIFTED from true local noon by the vessel's motion
(net motion relative to the Sun's GP) and THEREFORE the curve is NOT
SYMMETRICAL ABOUT LOCAL NOON. There is no need for you to go on further
about this point. You have not found some flaw in other people's
understanding.

And you wrote:
"As an example, let's take the ship approaching the Clyde in Winter, that I
tried to get Frank to consider as a noon-longitude exercise (but he
ducked)."

Hmmm. That's not true, George. I did not "duck" your example. I answered it.

And you wrote:
"But not so the difference in the timing of that central value, to anyone
trying to use Frank's proposed method to find his longitude. The correction
that has to be made for the ship's south-going speed of 10 knots is all of 5
minutes of time, or 1.25 degrees of longitude, which has to accommodate the
speed of the ship, any tidal current and any declination changes in the Sun
position."

The great part, George, is that you JUST DEMONSTRATED how easy it is to make
the correction. Run your example in reverse: take the observed altitudes,
SUBTRACT the effect of the vessel's motion, and then you have the altitudes
that would be observed by a motionless observer. It is indeed an EASY
process. We could teach a child to do it. The arithmetic and the concepts
are that simple.

You worried that we have to include not just the speed of the ship through
the water (which is readily available in any modern vessel) but also
currents and the changing declination of the Sun. For the latter, that is
also easy to include. Reading out the hourly rate of change of the Sun's
declination is probably one of the simplest things you could do with an
almanac. If you don't like the idea of providing the navigator with a
full-blown almanac, a very short table can give the rate of change of
declination (we're going to need a table anyway for the equation of time, so
might as well add a short table of rate of change of dec). As for currents,
they are typically less than one knot in open ocean. If you're in the Gulf
Stream or the Kuroshio or one of the other big, fast currents peeling off
from the eastern coasts of large landmasses, then you may be in trouble
since currents can exceed four knots frequently. BUT the point you're
missing is that this applies to ALL running fixes --not just this method. If
a hide-bound, traditionalist navigator chose to eschew this method of
finding longitude at noon, his alternative would be to cross a couple of
standard Sun LOPs separated by, let's say, four hours. And there will be an
error if there are unknown currents: a four knot current after four hours
would lead to a position error of 16 n.m. in a STANDARD running fix, if that
current is not taken into account. This is a property of ALL running fixes,
not a flaw with the particular method under discussion here.

And you added:
"With modern shipping commonly travelling at over 25 knots, it's
obvious what an enormous correction this must be, and how precisely it would
have to be made."

Well, heck, George, why stop at 25 knots? Let's imagine some advanced
watercraft zipping along at 60 knots. And let's put it up at latitude 60
north in January, too. I trust that these would meet even your requirements
for an extreme case, yes? Under those circumstances, the offset in noon
latitude would be about 15 nautical miles. Definitely worth correcting. And
the offset in the time of maximum altitude would be about 30 minutes
 --corresponding to 450 minutes of longitude or 225 n.m. at that latitude.
Now if we stupidly acted as if these sights were taken by an observer at
rest, we would be in deep trouble. But if we correct them, as we must for a
running fix, then we can get excellent results. Supposing we have a 1 knot
error in the net speed, how big do you suppose the error in latitude and
longitude would be, George? Have you tried setting this up as a simulation??

And George, you concluded:
"And yet Frank is claiming that he can derive the central moment of that
altitude curve, and then make that correction"

First, let's be clear that I recommend doing that in reverse order: correct
the raw sights for motion and THEN find the axis of symmetry of the curve
(by plotting them on graph paper, folding the paper in half, and lining the
points up as best as possible to make half a parabola). If you find the axis
first, it's considerably more difficult computationally which defeats much
of the advantage of using this method.

And also:
"to provide an overall error in the whole process of no more than 5 miles in
longitude, which corresponds to 35 seconds of time."

Careful there, George. Please do not mis-quote me. I did NOT say "no more
than 5 miles." And I think you know that.

And George, you concluded:
"No wonder he is reluctant to disclose the details."

George? Did you read my last post? As I said the other day, I wrote up a
fairly detailed account of this for the group way back in June, 2005. I
provided a link to it a couple of days ago. I don't fault you for not
remembering a post from that long ago, but I did assume that the general
method would be memorable. This time, I will paste the full text in:

The post was titled "Latitude AND Longitude by "Noon Sun"":
"First things first: I've put the phrase "Noon Sun" in quotes here because
the set of sights required for this system goes a little beyond the standard
procedure for shooting the Noon Sun for latitude only.

This short method of celestial navigation will get you latitude and
longitude to about +/-2 miles and +/-5 miles respectively --more than
adequate for any conceivable modern practical purpose. You can cross oceans
safely and reliably for years on end using this technique if it suits you to
do so. Its enormous advantage is simplicity. It's easy to teach, easy to
demonstrate, easy to learn, and also easy to re-learn if necessary. I
mention this because most people who are learning celestial navigation today
will quickly forget it. What's the point of learning something if you can't
reconstruct your knowledge of it quickly when and if the need actually
arises to use it? It's tough to resurrect an understanding of the tools of
standard celestial navigation on short notice, but easy with this lat/lon at
noon method. Additionally, this method does not require learning all the
details of using a Nautical Almanac (you don't need one at all --only a
short table of declination and equation of time, possibly graphed as an
"analemma") and it needs no cumbersome sight reduction tables.

Here's how it's done:

Start 20 or 30 minutes before estimated local noon. Shoot the Sun's altitude
with your sextant every five or ten minutes (or more often if you're so
inclined) and record the altitudes and times by your watch (true GMT).
Continue shooting until 20 or 30 minutes after local noon. [note the
difference from a noon latitude sight --we're recording sights leading up to
and following noon-- usually these are thrown away]

Next you need to correct for your speed towards or away from the Sun. For
example, if we're sailing south and the Sun is to the south of us, then each
altitude that we have measured will be a little higher as we get closer to
the latitude where the Sun is straight up. We need to 'back out' this effect
so that the data can be used to get a fix at a specific point and time. This
isn't hard. First, we need the fraction of our speed that is in the
north-south direction. If I'm sailing SW at 10 knots, then the portion
southbound (in the Sun's direction) is about 7.1 knots. You can get this
fraction by simple plotting or an easy calculation. Next we need the Sun's
speed. The position where the Sun is straight overhead is moving north in
spring, stops around June 21, then heads south in fall, bottoming out around
December 21 (season names are northern hemisphere biased here). It is
sufficient for the purposes of this method to say that the Sun's speed is 1
knot northbound in late winter through mid spring, 1 knot southbound from
late summer through mid autumn, and 0 for a month or two around both
solstices (it's easy to prepare a monthly table if you want a little more
accuracy). Add these speeds up to find out how much you're moving towards or
away from the Sun. If you're moving towards the Sun, then for every six
minutes away from noon, add 0.1 minutes of arc for every knot of speed to
the altitudes before noon and subtract 0.1 minutes of arc for every knot of
speed to the altitudes after noon. Reverse the rules if you're moving away
from the Sun. Spelled out verbally like this, this speed correction can
sound tedious but the concept is really very simple and it's very easy to
do. [Incidentally, George Huxtable deserves credit for emphasizing the
importance of dealing with this issue (although I don't think he ever
spelled out how to do it)]

Now graph the altitudes (use proper graph paper here if at all possible):
Sun's altitude on the y-axis versus GMT on the x-axis. The size of the graph
should be roughly square, maybe 6 inches by 6 inches so that you can clearly
see the rise and fall of altitude. For longitude, you will need to determine
the axis of symmetry of the parabolic arch of points that you've plotted.
There is a simple way to do this: make an eyeball estimate of the center and
lightly fold the graph paper in half along this vertical (don't "hard
crease" the fold yet). Now hold it up to the light. You can see the data
points preceding noon superimposed over the data points following noon which
are visible through the paper. Slide the paper back and forth until all of
the points, before and after, make the best possible smooth arch (half a
parabola). Now crease the paper. Unfold and the crease line will mark the
center of symmetry of the measured points with considerable accuracy.
Reading down along this crease to the x-axis, you can now read off the GMT
of Local Apparent Noon. Reading back up the crease to the data, you can pick
off the Sun's maximum noon altitude (which is probably already recorded but
if you missed the exact moment of LAN you can get it this way).

Next we need two pieces of almanac data: the Sun's declination for this
approximate GMT on this date and the Equation of Time for the same date and
time. You do NOT need a current Nautical Almanac for this. The exact value
of declination and Equation of Time varies in a four-year cycle depending on
whether this year is a leap year or the first, second, or third year after.
So we don't need an almanac for this. A simple table will do (where to get
one? Today, they're very easy to generate on-the-fly... or you could use an
old Nautical Almanac... or you could also use an analemma drawn on a
sufficiently large scale).

Apply the Equation of Time to the GMT of Local Apparent Noon that you found
above. You now have the Local Mean Time at LAN, and you already know the
Greenwich Mean Time. The difference between those two times is your
longitude. Convert this to degrees at the rate of 1 degree of longitude for
every four minutes of time difference. Done. We've got our longitude.

Now for latitude. Notice that we didn't correct any of our altitudes for
index correction or dip or refraction or the Sun's semi-diameter. These
corrections are totally unnecessary for the longitude determination. But we
need them for latitude. Take the Sun's altitude at the time of LAN (read off
the "crease" or actually observed by watching the Sun "hang" at the moment
of LAN). Correct it for index correction, dip, refraction and semi-diameter
as usual. This gives you the Sun's corrected observed altitude. Subtract
from 90 degrees. This "noon zenith distance" tells us how many degrees and
minutes we are away from the latitude where the Sun is straight up. The
latitude where the Sun is straight is, by definition, the "declination" that
we have looked up previously from our tables. So if the Sun is north of us
at noon, then we are south of the Sun's declination (latitude) by exactly
the number of degrees and minutes in the noon zenith distance. If the Sun is
south of us at noon, then we are north of the Sun's declination by the same
amount. A simple addition or subtraction yields the required latitude. Done.

We've spent about ten minutes making and recording observations of the Sun's
altitude over the course of 45 minutes to an hour, and reduced those
observations to get our latitude and longitude at noon with about five
minutes of paperwork. Not bad!

Again, the overwhelming advantage of this "short celestial" is that it can
be taught easily, learned quickly, and RE-learned quickly on the spot if
necessary. An additional advantage is that it requires an absolute minimum
of materials. You need a sextant (metal if at all possible, but plastic will
do), a decent, cheap watch or small clock, tables of refraction and dip (one
sheet of paper), a four-year revolving almanac of the Sun's declination and
equation of time (another sheet or two of paper), and some graph paper and a
pencil. You could even print out these (or equivalent) instructions and
throw everything in the case with your sextant.

As for disadvantages, they really depend on the student and his or her
expectations. What is it that we want to do with celestial navigation? Why
study any method? And for a thousand students, you will get a thousand
answers. The days are gone when celestial navigation was essential and fixed
curricula could be dictated for students to either take in their entirety or
leave. This field has moved on to the stage of "a la carte" learning. It can
be a pain in the neck for instructors accustomed to doing things the same
way year after year but it's a real liberation for students and possibly
also for more creative teachers and "information publishers".

Please note: there are some details that I would change and some additions
and amplifications that I would make if I were re-writing the above today.
I've quoted it unchanged for reference.


 -FER



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